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February  2016, 36(2): 971-980. doi: 10.3934/dcds.2016.36.971

Topological degree method for the rotationally symmetric $L_p$-Minkowski problem

1. 

Department of Applied Mathematics, Zhejiang University of Technology, Hangzhou 310023, China

2. 

Department of Mathematical Sciences, Tsinghua University, Beijing 100084

Received  October 2014 Revised  February 2015 Published  August 2015

Consider the existence of rotationally symmetric solutions to the $L_p$-Minkowski problem for $p=-n-1$. Recently a sufficient condition was obtained for the existence via the variational method and a blow-up analysis in [16]. In this paper we use a topological degree method to prove the same existence and show the result holds under a similar complementary sufficient condition. Moreover, by this degree method, we obtain the existence result in a perturbation case.
Citation: Jian Lu, Huaiyu Jian. Topological degree method for the rotationally symmetric $L_p$-Minkowski problem. Discrete and Continuous Dynamical Systems, 2016, 36 (2) : 971-980. doi: 10.3934/dcds.2016.36.971
References:
[1]

J. Ai, K.-S. Chou and J.-C. Wei, Self-similar solutions for the anisotropic affine curve shortening problem, Calc. Var. Partial Differential Equations, 13 (2001), 311-337. doi: 10.1007/s005260000075.

[2]

L. Alvarez, F. Guichard, P.-L. Lions and J.-M. Morel, Axioms and fundamental equations of image processing, Arch. Rational Mech. Anal., 123 (1993), 199-257. doi: 10.1007/BF00375127.

[3]

B. Andrews, Evolving convex curves, Calc. Var. Partial Differential Equations, 7 (1998), 315-371. doi: 10.1007/s005260050111.

[4]

J. Böröczky, E. Lutwak, D. Yang and G. Zhang, The logarithmic Minkowski problem, J. Amer. Math. Soc., 26 (2013), 831-852. doi: 10.1090/S0894-0347-2012-00741-3.

[5]

E. Calabi, Complete affine hyperspheres. I, in Symposia Mathematica, Vol. X (Convegno di Geometria Differenziale, INDAM, Rome, 1971), Academic Press, London, (1972), 19-38.

[6]

S.-Y. A. Chang, M. J. Gursky and P. C. Yang, The scalar curvature equation on $2$- and $3$-spheres, Calc. Var. Partial Differential Equations, 1 (1993), 205-229. doi: 10.1007/BF01191617.

[7]

W.-X. Chen, $L_p$ Minkowski problem with not necessarily positive data, Adv. Math., 201 (2006), 77-89. doi: 10.1016/j.aim.2004.11.007.

[8]

W.-X. Chen and C.-M. Li, A necessary and sufficient condition for the Nirenberg problem, Comm. Pure Appl. Math., 48 (1995), 657-667. doi: 10.1002/cpa.3160480606.

[9]

K.-S. Chou and X.-J. Wang, The $L_p$-Minkowski problem and the Minkowski problem in centroaffine geometry, Adv. Math., 205 (2006), 33-83. doi: 10.1016/j.aim.2005.07.004.

[10]

K.-S. Chou and X.-P. Zhu, The Curve Shortening Problem, Chapman & Hall/CRC, Boca Raton, FL, 2001. doi: 10.1201/9781420035704.

[11]

J.-B. Dou and M.-J. Zhu, The two dimensional $L_p$ Minkowski problem and nonlinear equations with negative exponents, Adv. Math., 230 (2012), 1209-1221. doi: 10.1016/j.aim.2012.02.027.

[12]

M. Ji, On positive scalar curvature on $S^2$, Calc. Var. Partial Differential Equations, 19 (2004), 165-182. doi: 10.1007/s00526-003-0214-0.

[13]

H.-Y. Jian and X.-J. Wang, Bernsterin theorem and regularity for a class of Monge Ampère equations, J. Diff. Geom., 93 (2013), 431-469.

[14]

M.-Y. Jiang, L.-P. Wang and J.-C. Wei, $2\pi$-periodic self-similar solutions for the anisotropic affine curve shortening problem, Calc. Var. Partial Differential Equations, 41 (2011), 535-565. doi: 10.1007/s00526-010-0375-6.

[15]

Y.-Y. Li, Prescribing scalar curvature on $S^n$ and related problems. I, J. Differential Equations, 120 (1995), 319-410. doi: 10.1006/jdeq.1995.1115.

[16]

J. Lu and X.-J. Wang, Rotationally symmetric solutions to the $L_p$-Minkowski problem, J. Differential Equations, 254 (2013), 983-1005. doi: 10.1016/j.jde.2012.10.008.

[17]

E. Lutwak, The Brunn-Minkowski-Firey theory. I. Mixed volumes and the Minkowski problem, J. Differential Geom., 38 (1993), 131-150.

[18]

E. Lutwak, D. Yang and G. Zhang, On the $L_p$-Minkowski problem, Trans. Amer. Math. Soc., 356 (2004), 4359-4370. doi: 10.1090/S0002-9947-03-03403-2.

[19]

E. Lutwak and G. Zhang, Blaschke-Santaló inequalities, J. Diff. Geom., 47 (1997), 1-16.

[20]

R. Schoen and D. Zhang, Prescribed scalar curvature on the $n$-sphere, Calc. Var. Partial Differential Equations, 4 (1996), 1-25. doi: 10.1007/BF01322307.

[21]

G. Szego, Orthogonal Polynomials, American Mathematical Society, Providence, R.I., fourth edition, 1975.

[22]

V. Umanskiy, On solvability of two-dimensional $L_p$-Minkowski problem, Adv. Math., 180 (2003), 176-186. doi: 10.1016/S0001-8708(02)00101-9.

[23]

G. Zhu, The logarithmic Minkowski problem for polytopes, Adv. Math., 262 (2014), 909-931. doi: 10.1016/j.aim.2014.06.004.

[24]

G. Zhu, The centro-affine Minkowski problem for polytopes, J. Differential Geom., 101 (2015), 159-174.

show all references

References:
[1]

J. Ai, K.-S. Chou and J.-C. Wei, Self-similar solutions for the anisotropic affine curve shortening problem, Calc. Var. Partial Differential Equations, 13 (2001), 311-337. doi: 10.1007/s005260000075.

[2]

L. Alvarez, F. Guichard, P.-L. Lions and J.-M. Morel, Axioms and fundamental equations of image processing, Arch. Rational Mech. Anal., 123 (1993), 199-257. doi: 10.1007/BF00375127.

[3]

B. Andrews, Evolving convex curves, Calc. Var. Partial Differential Equations, 7 (1998), 315-371. doi: 10.1007/s005260050111.

[4]

J. Böröczky, E. Lutwak, D. Yang and G. Zhang, The logarithmic Minkowski problem, J. Amer. Math. Soc., 26 (2013), 831-852. doi: 10.1090/S0894-0347-2012-00741-3.

[5]

E. Calabi, Complete affine hyperspheres. I, in Symposia Mathematica, Vol. X (Convegno di Geometria Differenziale, INDAM, Rome, 1971), Academic Press, London, (1972), 19-38.

[6]

S.-Y. A. Chang, M. J. Gursky and P. C. Yang, The scalar curvature equation on $2$- and $3$-spheres, Calc. Var. Partial Differential Equations, 1 (1993), 205-229. doi: 10.1007/BF01191617.

[7]

W.-X. Chen, $L_p$ Minkowski problem with not necessarily positive data, Adv. Math., 201 (2006), 77-89. doi: 10.1016/j.aim.2004.11.007.

[8]

W.-X. Chen and C.-M. Li, A necessary and sufficient condition for the Nirenberg problem, Comm. Pure Appl. Math., 48 (1995), 657-667. doi: 10.1002/cpa.3160480606.

[9]

K.-S. Chou and X.-J. Wang, The $L_p$-Minkowski problem and the Minkowski problem in centroaffine geometry, Adv. Math., 205 (2006), 33-83. doi: 10.1016/j.aim.2005.07.004.

[10]

K.-S. Chou and X.-P. Zhu, The Curve Shortening Problem, Chapman & Hall/CRC, Boca Raton, FL, 2001. doi: 10.1201/9781420035704.

[11]

J.-B. Dou and M.-J. Zhu, The two dimensional $L_p$ Minkowski problem and nonlinear equations with negative exponents, Adv. Math., 230 (2012), 1209-1221. doi: 10.1016/j.aim.2012.02.027.

[12]

M. Ji, On positive scalar curvature on $S^2$, Calc. Var. Partial Differential Equations, 19 (2004), 165-182. doi: 10.1007/s00526-003-0214-0.

[13]

H.-Y. Jian and X.-J. Wang, Bernsterin theorem and regularity for a class of Monge Ampère equations, J. Diff. Geom., 93 (2013), 431-469.

[14]

M.-Y. Jiang, L.-P. Wang and J.-C. Wei, $2\pi$-periodic self-similar solutions for the anisotropic affine curve shortening problem, Calc. Var. Partial Differential Equations, 41 (2011), 535-565. doi: 10.1007/s00526-010-0375-6.

[15]

Y.-Y. Li, Prescribing scalar curvature on $S^n$ and related problems. I, J. Differential Equations, 120 (1995), 319-410. doi: 10.1006/jdeq.1995.1115.

[16]

J. Lu and X.-J. Wang, Rotationally symmetric solutions to the $L_p$-Minkowski problem, J. Differential Equations, 254 (2013), 983-1005. doi: 10.1016/j.jde.2012.10.008.

[17]

E. Lutwak, The Brunn-Minkowski-Firey theory. I. Mixed volumes and the Minkowski problem, J. Differential Geom., 38 (1993), 131-150.

[18]

E. Lutwak, D. Yang and G. Zhang, On the $L_p$-Minkowski problem, Trans. Amer. Math. Soc., 356 (2004), 4359-4370. doi: 10.1090/S0002-9947-03-03403-2.

[19]

E. Lutwak and G. Zhang, Blaschke-Santaló inequalities, J. Diff. Geom., 47 (1997), 1-16.

[20]

R. Schoen and D. Zhang, Prescribed scalar curvature on the $n$-sphere, Calc. Var. Partial Differential Equations, 4 (1996), 1-25. doi: 10.1007/BF01322307.

[21]

G. Szego, Orthogonal Polynomials, American Mathematical Society, Providence, R.I., fourth edition, 1975.

[22]

V. Umanskiy, On solvability of two-dimensional $L_p$-Minkowski problem, Adv. Math., 180 (2003), 176-186. doi: 10.1016/S0001-8708(02)00101-9.

[23]

G. Zhu, The logarithmic Minkowski problem for polytopes, Adv. Math., 262 (2014), 909-931. doi: 10.1016/j.aim.2014.06.004.

[24]

G. Zhu, The centro-affine Minkowski problem for polytopes, J. Differential Geom., 101 (2015), 159-174.

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